CUSAT CAT 2013 Mathematics Question Paper

IIIIIIIHI

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TEST BOOKLET No.

MATHEMATICS

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Time: 2 Hours

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TEST FOR POST GRADUATE PROGRAMMES

m593 o c Maximum Marks: 450

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INSTRUCTIONS TO CANDIDATES

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You are provided with a Test Booklet and an Optical Mark Reader (OMR) Answer Sheet to mark your responses. Do not soil the Answer Sheet. Read carefully all the instructions given on the Answer Sheet.

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Write your Roll Number in the space provided on the top of this page.

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Also write your Roll Number, Test Code, and Test Subject in the columns provided for the same on the Answer Sheet. Darken the appropriate bubbles with a Ball Point Pen.

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The paper consists of I50 objective type questions. All questions carry equal marks.

Each question has four alternative responses marked A, B, C and D and you have to darken the bubble fully by a Ball Point Pen corresponding to the correct response as indicated in the example shown on the Answer Sheet.

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Each correct answer carries 3 marks and each wrong answer cznrics 1 minus mark.

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Space for rough work is provided at the end of this Test Booklet.

You should return the Answer Sheet to the Invigilator before you leave the examination hall. However, you can retain the Test Booklet.

Every precaution has been taken to avoid errors in the Test Booklet. In the event of any such

unforeseen happenings, the same may be brought to the notice of the Observer/Chief Superintendent in writing. Suitable remedial measures will be taken at the time of evaluation, if necessary.

1 MIIMIIIEII MATHEMATICS

The linear function f(x) = mx + b where m at 0 is (A) continuous everywhere (C) continuous when m > 0

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(B) not continuous (D) None of the above

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The absolute extreme values of f(x) = x3 — 3x + 2 on [0,2} are

m (A) 0,4 (B) -1,4 e (C) 4,2 (D) 0,3 s a l g

T he point on the parabola y = (x — 3)2 that is nearest the origin

(A) (2.'"1) (13) a(2, 1) . *1) (C) ('2. 1) (D) (*2, .t——>c/3 2 lim X3 sin

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(A) 1s (B) -00 (C) 00 i(D) None of the above m d

The last digit of 2007’°°3 is

U:

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(A) 9 (D) (B) 17 .a 3 (C) w

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When an article is sold at b rupees and a paise after purchasing it at a rupees and b paise, the loss incurred is c rupees. Then the correct relation among a, b, c is

a+b c = 0.99 D

(A) c= 0.99 (B) - — 0.99

((-2) ( ) c C

(C)

2 IIIIIIIIII 852 digits are used to number the pages of a book consecutively from page 1. The number of pages in the book is

(A) 284 (C) 320

(B) 316 (D) 351

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The number of zeros used in writing natural numbers from 1 to 200 is

(A) 21 (C) 31

(B) 25 (D) 41

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There will be a CAT-2012 (Common Admission Test) in the

a . s (A) 508ion(B) s is m d (B) 46 a (C) . 57 w

University. Let CAT consist of three distinct positive integers C, A, T such that the product C x A x T = 2012. The largest possible value ofthe sum C + A + T is 1006

(C) 1009

(D) 2014

Bhaskar distributes some chocolates to four boys in the ratio 1/3 1/4

1/5 1/6. The minimum number of chocolates that Bhaskar should haveis

(A) 23

(D) 60

11.

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In the following display ifl B I9 [D P515] , each letter represents a digit. If the sum of any three successive digits is 15, then the value of E is

(A) 0 (C) 7

(B) 2 (D) 8

3 lfllllflllll 12.

A rectangular sheet of metal, x cm b5/”y cm, has a square of side 2 cm

cut from each comer. The sheet is then bent to form a nay of depth

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2 cm . The volume (in cc) of the tray is

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(A) Z=(x-}’)()'*Z) (B) )9’?

(C) (x+y)z (D) z(x—2z)(y—2z) 13.

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If a circle and a square have the same perimeter, then the area of the

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circle is

(A) smaller

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(B) 71; times the area of the square

(C) equal to the square

(D) larger 14.

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If a circle has radius 2 cm and a square is inscribed in the circle, then the area of the square (in sq. cm) is

(A) 2 (C) 4

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(B) 2\/E (D) 8

If 3”=4,4”= 5, 5"=6, 6’=7, 7“-"8 and 8"=9,thenthevalueof

15.

a . 9 w(A) (C) 2

the product ‘p q r s t 21’ is

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16.

(B) 8 (D) 1

If A is the set of natural numbers and B is the set of integers, the

symmetric difference of A and B is

(A) set of natural numbers (B) (C) {0,~1,~2,...}

set of integers

(D) {~1,#2,—3,...}

17.

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H‘ A and B have 50 elements each and A ~ B has 40 elements, then

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(A) 80 (D) (B)100 90 (C) 95

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18. 19 boys tum out for baseball. Of these 1_l are wearing base ball shirts and 14 are weafing baseball pants. There are no boys without one or

a l (A) (B) 53ag (C) 53 (D) . s n (A) X L) Yio(B) Y (C) X’s(D) ¢ is m (A) 3 (B) 6 d (C) 9 (D) 18 .a p S q + r w 21. w The value of q 5 r+ p is rSp+q w other. The number of boys wearing full uniform is

19' If X and 1’ are two sets, then Xn(y-U X)C equals

20_ Sets A and B have 3 and 6 elements each. The minimum number of elements in Av B can be

(A) 5+p+q+r

(B) s(p+q+r)

((3) p + q + r

(D) 0

5 llllflllllfll

2 S 25

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I23

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22. If 2 x 3 :0, then .1" is equal to

345

m A ~-~ B -­ e Oand b > 0 is

31.

(A) independent of a (B) dependent on a

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(C) independent ofa when a > and dependent on a when a <

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(D) None of the above 32.

The set {1,sinx,cosx,sin2x.cos2x. ...} is orthogonal on

a 71' T? “H 11‘ l (A) [51 (B) (7.; g a (C) I—~n.n]s.(D)

If

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(A) converges to 21:

(C) diverges

,,, 1

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(B) converges (D) None of the above

,,=2;E‘-E-l—)—l; converges for

(A) .a p>1 (B) p>[§) and p>1

w(C) p 2 1 (D) None of the above

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35.

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Let 2 an be conditionally convergent.

'_'(1n < O

Define{an p,,= >0. anand 2< 0‘ 1 n___ 0 ifif an 0 q,,={ 0 _.fa

Then

(A) )_:p,, converges (C) Both diverges

(B) Eqn converges (D) Both converges

8 IIIIIIIIII .0 c—1)"cx—2)".

36.

Interval of convergence of X "*9 (n~I-D33"

(A) [*1.1] (C) V1.5}

(B) [—1.2IU[3.5] (D) None of the above

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m e 2 -7 -22 25 4a-22 s W E'§'('?'‘‘:T’) 0'” a'3~(:‘a';'*;') l (C) E ,(~1;Z,3E) (D) None of the above g a 22 X2 y2 ' . §'F"E=1” s n o i s is m d a . w -2; 2 2­

37.

Curvature and centre of cuwature at p(2,1) on the curve r(!)=0

38.

(A) elliptic hyperholoid of two sheets (B) hyperboloid of two sheets

(C) elliptic cone (D) None of the above

39.

iim 23.3’

(-*-)')—‘("«°} 3:2 + y2

(A) exists (C) exists when .ry=l

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(B) does not exist (D) None of the above

Ifz-—x+y, xax-F ya)’­

m)z (m 3 (C) (D) None of the above \ 0 and y > 0

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80.

6(m') If .t- = r C030 and y *4 r3:'n9 , then aha) is given by

(A) r (B) -r (C) 1 (D) 1 RI

16 IIIIIIIIII 81.

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f0‘(1 — x)"'-1 x’‘‘1 dx is

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(A) divergent (B) convergent for all the values of m and 1:

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(C) converges for m >0 and n > 0 (D) converges for m < 0 and n < 0 82.

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E [C (x dy -- 32 dx) gives the

(A) volume enclosed by the curve (B) area enclosed by the curve (C) length of the curve (D) surface area of the curve

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The Laplace lransfonn of the function is

83.

(A) tan” 5

(B) cot” S (D) cot’1(s + 1)

i 1 co co’ m dm2 1 to a . 0 (B) 11 (A) w(C) c + id is defined only when

(A) a=0andc=0 (C) a=0andd=0 115.

(B) c=0andd=0 (D) b=0andd=0

The point of intersection of the lines given by 1'2 -Say +4y’ +x+2y—2 = O is

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(A) (L2) (C) (2.1)

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(B) (-1.2) (D) (0.0)

a . s (A) 32 (B) 15 n (c) 64 (D) 2:: o i s is (D) x+4a=0 m d a . w

116.

117.

The area enclosed by the curves y = 4x3 and y=16x is

Two peipendicular tangents to y’ = ‘lax always intersect on the line (A) .1’ ——a = 0

(B) x+a==0

(C) .1‘ + 2n = 0

118.

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119.

The point which is equidistant from the points (0,0,0), (2,0,0), (0,2,0) and (?.,2,2) is

(A) (1,o,1)

(B) (0.13)

(C) (1.1.—1)

(D) (1.1,1)

The volume of the parallelepiped whose edges are represented by the vectors 1'+_;', j+l(, k-H’ is

(A) 2 (C) 1

(B) 0 (D) 6

23 IIIIWIHIIWIII 120.

Let {(2) = cos 2: — $3 for non-zeroz e c and f(0) = 0, then _f_(z) has a zero at (A) z = 0 of order 1

(C) 2 = I of order 3 I21.

(B) z=0 of order 2 (D) z=I of order 2

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If ft’ , F and 3 are three mutually perpendicular unit vectors, then

|a+B+a|e

122.

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(A) «E (B) 3 (C) 2 (D) 0 gla a . s n o i s is m d a . w

Let the characteristic equation of the matrix M be 23 - X - 1 = 0, then

(A) M " does not exist (3) M "' exists but can't determine from the data

(C) M " = M +1 (D) M“ = M ~1

123.

The maximum magnitude of the directional derivative for the surface x2 + xy + yz 2: 9 at the point (1 ,2,3) is along the direction (A) 1' + J’ + 72'

(C) ‘H 2]’-t k

124.

(B) ?+2;"’+4E' (D) ?+2}“+2k

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The number of linearly independent eigen vectors of the matn'x

w 0030“ 2100. 001<

(M1 (Q3 (13)? (D)4

24 IIIIIII 125.

The function f(z) = ,/lxyl is (B) nowhere analytic (D) not continuous

(A) cveiywhere analytic (C) not analytic at origin 126.

11

. x . m. e s A) J5 -B ­ ((C) ( ) 2 a 1 (D) 2l

x and y are two numbers and x < y. If x’ + y’ equals two times five and xy equals two tunes two, then was )3

g a .

127.

If I00 = .1‘ (mod 7), then least positive value of x is

128.

The integral of I1+e ex“' dx is

s (A) 7 (B) n 4 (C) 3 o(D) 2

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i (A)m 32-‘ +C (B) tan” (e")+C (C) sec" (e“)+ C (D) log e’'‘ + C d .a

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129.

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The value of Itanxdx is

(A) logsinx (B) logcosx (C) —logcos.x' (D) logsecx 3 < x < 5 is equivalent to

(A) x = 4 (B) 3:

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-D­

(C) [.1-—4[l

25 llflillllllllll 131.

The equation x2 + y2 — 2.ry~l == 0 represents

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(A) a circle (B) two perpendicular straight lines (C) two parallel straight lines

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(D) hyperbola I32.

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If A: {l,2,3,4,5}, (hen the number of proper subsets of A is

(B) 32 (D) S

(A) 120 (C) 31 133.

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The value of ]‘(x2 *1) (Ix is (x2 +1)

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(A) 108 (1 4' 3'4) (B) .1‘ — 2 tan" x

i s is m (A) 2 (13) 1 (C) '/I (D) 0 d .a w

(C) .r + 2 tan" x (D) tan" x’

I34.

135.

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136.

The value of I.r!xldA' is

If .r=a(cos9+6sin9) and }':=a(sin6-—6cos(9), then ii): = .1

(A) sin6 (C) tan9

(B) 1x (D) cos6

If every element of a group G is its own inverse, then G is

(A) ring

(B) field

(C) cyclic group

(D) abelian group

IIIIIIIEI 137. For the differential equation of all straight lines passing through the

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. . dy .

ongm —— as (Ix

. (A) (B) — (C) x+ y (D) 2(x— y) m e s a l

133. If A ={1,3,4,6} and B={l,9,16,36} and a mapping fis defined by f (a) :1); are A and b 6B and b=a’,thcu the mapping f is

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(A) only injective mapping (B) only surjective mapping

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(C) bijective mapping (D) None of these mapping 139.

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If f:R —-> R is defined by f(x) -—- 3:’, then f(x) is

(A) one-one

(B) onto

s i 140. If f(x) = m (xfil) , then is equal to d a . -.1’ (B) (A) w(0) —f(x) (D) (x) w w (C) both one-one and onto

(D) neither one-one nor onto

(x-i-1)

-1

141. The focus of the parabola yz —~ 8.1‘ — 32 2 0 is at the point

(A) (~10) (B) (0.4) (C) (0,2)

(D) (4,0)

27 IIMIIHIIIIII 142.

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A box contains 6 white and 4 black balls. If 5 balls are drawn at

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random from this box, the probability of getting exactly 3 white and 2 black balls is

_.2B( 3..6 (A)

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(C) -11% (D) None of the above

a l (A) sinx cosh y (B) (sin g1: sinh y (C) cosx cosh y (D)a cosx sinh y . s n (A)ssio(B) (C) i (D) m d a .

143.

144.

The imaginzuy part of sin (x + iy) is

A number of 5 digits is formed with digits 1,2,3,4,S without repetition. The probability that it is a number divisible by 4 is

L/x[.:>. U1]t~J

145.

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A student has to select 3 subjects out of 6 subjects - Mathematics,

English, Hindi, Science, I-Iistoiy and IT. If the student has chosen IT, the probability of choosing Mathematics is

(A) (B) (C) (D)